I have a problem; Suppose X1,X2,X3,X4 are independent with common mean 1 and com

Question

I have a problem;

Suppose X1,X2,X3,X4 are independent with common mean 1 and commonvariance 2. Compute the Cov(X1+X2 , X2+X3).

I'm a bit confused on this one, if assume Y=X1+X2 and Z=X2+X3 andCov(U,V) = E[UV] - E[U]E[V]. Usually i'd assume if Z = X3 +X4 the two variables Y and Z have nothing in common thus you cansplit the E[UV] into E[U]E[V] and the covariance (naturally) is0. However with the shared inner variable how would one solvethis? This was my approach but i'm not sure if its right;

Cov(Y,Z) = E[(X1+X2)(X2+X3)] - E[X1+X2]E[X2+X3]
= E[X1X2 + X1X3 + X2^2 + X2X3] - E[X1 + X2]E[X2 + X3]
= E[X1X2] + E[X1X3] + E[X2^2] + E[X2X3] - (E[X1] +E[X2])(E[X2] + E[X3])
= E[X1X2] + E[X1X3] + E[X2^2] + E[X2X3] - E[X1]E[X2] - E[X1]E[X3] -E[X2]^2 - E[X2]E[X3]
=E[X1]E[X2] + E[X1]E[X3] + E[X2^2] + E[X2]E[X3] - E[X1]E[X2] -E[X1]E[X3] - E[X2]E[X2] - E[X2]E[X3]
= E[X2^2] - (E[X2])^2

or Var(X2) = 2.

Explanation / Answer

It is simpler to use this approach: Uing the fact that Cov(X,Y) = E(XY) - E(X)E(Y), we can derive that Cov(X+U, Y+V) =Cov(X,Y)+Cov(X,V)+Cov(U,Y)+Cov(U,V) Therefore, Cov(X1+X2 , X2+X3) = Cov(X1, X2) + Cov(X1, X3) + Cov(X2, X2) +Cov(X2, X3) = 0 + 0 + Var(X2) + 0 = 2 Since X1 and X2 are indep, X1 and X3 are indep, X2 and X3 areindep.

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